Advection surface-flux balance controls the seasonal steric sea level amplitude

Along with the mean sea level rise due to climate change, the sea level exhibits natural variations at a large number of different time scales. One of the most important is the one linked with the seasonal cycle. In the Northern Hemisphere winter, the sea level is as much as 20 cm below its summer values in some locations. It is customary to associate these variations with the seasonal cycle of the sea surface net heat flux which drives an upper-ocean thermal expansion creating a positive steric sea level anomaly. Here, using a novel framework based on steric sea level variance budget applied to observations and to the Estimating the Circulation and Climate of the Ocean state estimate, we demonstrate that the steric sea level seasonal cycle amplitude results from a balance between the seasonal sea surface net heat flux and the oceanic advective processes. Moreover, for up to 50% of the ocean surface, surface heat fluxes act to damp the seasonal steric sea level cycle amplitude, which is instead forced by oceanic advection processes. We also show that eddies play an important role in damping the steric sea level seasonal cycle. Our study contributes to a better understanding of the steric sea level mechanisms which is crucial to ensure accurate and reliable climate projections.


Figure S1
In Figure S1 of this supplementary file, we show the time mean of the product between the seasonal cycle of Sea Surface Temperature (SST) and the seasonal cycle of the net heat flux i.e.SST Q ρ0Cp where ρ 0 is a reference density with ρ 0 = 1029 kg m −3 and C p = 4000 J kg −1 K −1 the specific heat of seawater.The seasonal cycle of the SST, used in Figure (S1) is obtained from the NOAA OI SST v2 dataset [1], which is available on a 1/4 • global grid from 1981.We first select the period 1993-2014, corresponding to ECCO v4r3, remove the trends, the time mean, and compute the seasonal cycle from the monthly time series.The net heat flux is computed from the sum of the net short wave, net long wave, latent and sensible heat fluxes, given by ERA5 [2] on a 1 /4°grid.Figure S1 shows that SST Q ρ0Cp is positive everywhere, demonstrating that on seasonal time scales the net heat flux acts to increase the amplitude of the seasonal cycle of SST.This is in strong contrast to the results obtained for the seasonal cycle of steric sea level (Figure 1 of the main manuscript), which is negative over about half of the ocean surface.

Figure S2
In Figure S2 of this supplementary file, the buoyancy flux term of the SSL seasonal cycle variance budget (VAR flu ) is decomposed into a part linked with the net heat flux (VAR flu θ ) and a part linked with freshwater fluxes (VAR flu S ), that is: VAR flu is almost everywhere controlled by the net heat flux variations (Fig. S2a).The freshwater fluxes term VAR flu S (Fig. S2b) has its largest magnitude at low latitudes in both hemispheres and can either be positive (a source of SSL seasonal cycle variance) or negative (a sink).When globally averaged, both terms are positive: 0.1 cm 2 yr −1 for VAR flu S and 0.8 cm 2 yr −1 for VAR flu θ , and confirm that VAR flu θ is the largest contributor to VAR flu which has a global average of 0.9 cm 2 yr −1 .

Figure S3
In Figure S3 we show the comparison between the seasonal cycle of the SSL time tendencies ( ∂η ∂t ) and the seasonal cycle of the term linked with the net heat flux ( αQ ρ0cp ).Results are averaged North of 30 • N in order to select only the Northern hemisphere phase and remove low latitudes where the phase is less clear (see Fig. 2a,b).Note that we have checked that results presented below also hold for the Southern hemisphere.The average is computed for regions where αQ ρ0cp > 0 (Fig. S3a), αQ ρ0cp < 0 (Fig. S3 b) and for regions where αQ ρ0cp > 10 cm 2 yr −1 (Fig. S3c), VAR flu < −10 cm 2 yr −1 (Fig. S3d).In this figure we see that ∂η ∂t occurs before αQ ρ0cp in regions where VAR flu > 0 (Fig. S3a) while it occurs after αQ ρ0cp in regions where VAR flu < 0 (Fig. S3b).This time lag of approximately 2 weeks becomes larger (up to 3 weeks) when the average is computed over regions where |VAR flu | > 10 cm 2 yr −1 .(Fig. S3c,d).This relatively small time lag is sufficient to induce a large SSL variance flux with magnitudes ∼ 30 cm 2 yr −1 (see Fig. 1 of the main manuscript).

Figure S4
In Figure S4 of this supplementary file, we compute the steric sea level variance budget using all frequencies of the ECCO state estimate.The budget equation is similar to equation (2) of the main manuscript (see also section Method): The only difference with Eq. ( 2) from the main manuscript is that instead of using the mean seasonal cycle of every term, we now use the detrended anomalies of each term.The superscript all in Eq. ( 2) indicates that each term contain all frequencies including interannual, annual and subannual.Fig. S4 shows each of the four terms of this variance budget.The results are very similar to the results obtained for the mean seasonal cycle (Fig. 3 a,b,c,d).The main balance is indeed between VAR all flu and VAR all adv and sources and sinks regions follows the same pattern as in Fig. 3.The main differences are located close to the equator in the Pacific ocean.The global average of each term is however larger when all frequencies are included.The fact that the variance budget terms are similar when all frequencies are included and when only the mean seasonal cycle is considered means that the SSL variance fluxes are dominated by the seasonal cycle.Moreover it shows that the methodology we use to extract the mean seasonal cycle (time mean of the monthly time series) is able to successfully extract the main characteristics of the seasonal cycle.

Figure S5
In Fig. S5, we show the southern and northern hemisphere averages of the budget for the seasonal variations of the SSL given by Eq. ( 14) of the main manuscript.In both hemispheres, the term associated with the surface buoyancy flux (mainly due to the net heat flux) is close to the seasonal trends of the SSL, but present a small time lag, of the order of a few weeks which implies a positive SSL variance flux of 5 cm 2 yr −1 in the NH and 3 cm 2 yr −1 in the SH.The advective term < adv > has a weaker amplitude than the < flu > term (∼5 cm yr −1 vs. ∼15 cm yr −1 in the NH), but is almost opposite to the hemispheric variations of < η > and acts to balance the effect of the surface buoyancy flux in both hemispheres.

Figure S6
In Fig. S6, we compute the term linked with the buoyancy fluxes in the SSL variance budget (VAR flu ) assuming that there is no density anomalies below the surface layer, that is: where h = 10 m and corresponds to the first layer of the ECCO V4r3 state estimate, and where flu ρ is the buoyancy forcing term of the density evolution equation (see section "Seasonal SSL variance budget" in Methods).VAR flu SURF is almost everywhere positive (Fig. S6), showing that the seasonal cycle density anomalies at the surface are in phase with the surface buoyancy flux seasonal cycle.

Figure S7
In Figure S7, we assess the differences between the seasonal variations of the steric sea level (η) and the seasonal variations of sea surface anomalies (SSA).SSA is corrected from the global mean sea level so that the difference between η and SSA is the manometric sea level, i.e.: SSA and η are obtained from the ECCO v4r3 state estimate for the period 1993-2014.The seasonal cycle is obtained by first removing the trend and then by computing the time average for each individual month resulting in a 12 points time series.Figure S7 shows the standard deviation of η (left panel) and the standard deviation of the difference between SSA and η.At low and mid latitudes the manometric component of the seasonal sea level is generally negligible compared to the steric sea level except in several semienclosed regions such as the Arabian sea or the Yellow sea.At high latitudes, particularly in the Arctic ocean, the standard deviation of the manometric sea level becomes important and cannot be neglected.

Figure S8
In Figure S8, we reproduce Figure 1b of the main manuscript i.e. η αQ ρ0Cp but using the net heat flux Q OAflux from the Woods Hole Oceanographic Institute Objectively Analyzed air-sea Fluxes for the global oceans (OAFlux) [3] instead of ERA-5 from ECMWF.OAFlux uses objective analysis to obtain optimal estimates of flux-related surface meteorology and then computes the global fluxes by using the state-of-the-art bulk flux parameterizations [4].The OAFlux net heat flux used here has a 1°horizontal resolution, is available over the period 1993-2009 and is interpolated on the AVISO 1 /4°grid.As before, the mean seasonal cycle is extracted from Q OAflux by first removing the trend and then by computing the time mean over each individual month.Figure S8 shows that using the OAflux net heat flux seasonal cycle extracted to compute η αQ ρ0Cp gives similar results to the one obtained from ERA-5.Differences (Fig. S8b) are almost everywhere one order of magnitude smaller than the original signal, largest values are found in eddy rich regions but this is likely due to the coarse 1°resolution of OAFlux compared to the 1 /4°of ERA-5.We conclude that our estimate of η αQ ρ0Cp is robust.Fig. S4 The SSL variance budget is similar to the SSL seasonal cycle variance budget.Same as Figure 3 a,b,c,d from the main manuscript except that the variance is computed from the monthly outputs, such that all frequencies are used (interannual, annual and subannual) to compute the budget and the seasonal cycle is not extracted.

Fig. S1
Fig. S1 The time mean of the product between the seasonal variations of net heat flux and SST is everywhere positive.SST is obtained from the NOAA OI SST v2 dataset and net heat flux from the ERA5 reanalysis.Units are in • C 2 days −1 .

Fig
Fig. S3 A non negligible time lag is found between the seasonal cycle of the SSL time tendencies ( ∂η ∂t ) and the seasonal cycle of the net heat flux term ( αQ ρ 0 cp ).The seasonal cycle of the net heat flux (orange lines) and time tendencies of SSL (blue lines) are averaged North of 30 • N in regions where VAR flu > 0 (panel a) and VAR flu < 0 (panel b) and in regions where VAR flu > 10 cm 2 yr −1 (panel c) and VAR flu < −10 cm 2 yr −1 (panel d).
Fig. S5 Hemispheric horizontal average of the seasonal SSL budget terms.Left panel: southern hemisphere, right panel: northern hemisphere.The red line represents the hemispheric variations of the SSL (in cm), the blue, orange and green lines respectively show the hemispheric horizontal average of the time tendencies of the SSL, buoyancy surface flux and advection.Diffusion is not shown because it is negligible.Budget terms are in cm yr −1 .

Fig
Fig.S6The seasonal cycle of buoyancy forcing is in phase with surface density anomalies almost everywhere.Term VAR flu SURF (in cm 2 yr −1 ) which is the same as VAR flu except that sub-surface density anomalies are set to zero (see text 1.6).

Fig. S7
Fig. S7 Seasonal steric sea level is well approximated by total sea level except at high latitudes and in several semi-enclosed seas.(a) standard deviation of the seasonal steric sea level (η in m) obtained in ECCO, (b) standard deviation of the difference between the total seasonal sea level anomalies (SSA − η in m) and the steric sea level.

Fig. S8 The estimate of η αQ ρ 0
Fig. S8 The estimate of η αQ ρ 0 Cp is robust to the choice of net heat flux dataset.Estimate (in cm 2 yr −1 ) using the net heat flux values derived from the OAFlux dataset (a) and differences between the ERA-5 and OAFlux estimates (b).